I've heard Udi can help with sleep issues. What specific examples does the Satcharita give?

Yes, Chapter 33 provides two distinct examples of Udi curing insomnia. The first story is about a friend of R. Hari Sitaram Dixit who had been unable to sleep for eight days, even with a doctor's medicine. After being given some of Maharaj's Udi and taking it for just one night, he returned the next day to report that he had slept very well throughout the night. The second anecdote mentioned in Chapter 33 concerns a child from Bandra who had not been able to sleep for two whole months. His father took him to Shirdi for Maharaj's Darshan, and that very night, the child got sleep, and the issue was resolved from then on.